On the limit cycles for a class of discontinuous piecewise cubic polynomial differential systems

On the limit cycles for a class of discontinuous piecewise cubic polynomial differential systems

Bo Huang

LMIB-School of Mathematical Sciences, Beihang University, Beijing, 100191, P. R. China Courant Institute of Mathematical Sciences, New York University, New York, 10012, USA

Received 4 October 2019, appeared 9 April 2020 Communicated by Armengol Gasull

Abstract. This paper presents new results on the bifurcation of medium and small limit cycles from the periodic orbits surrounding a cubic center or from the cubic center that have a rational first integral of degree 2 respectively, when they are perturbed inside the class of all discontinuous piecewise cubic polynomial differential systems with the straight line of discontinuityy=0.

We obtain that the maximum number of medium limit cycles that can bifurcate from the periodic orbits surrounding the cubic center is 9 using the first order averaging method, and the maximum number of small limit cycles that can appear in a Hopf bifurcation at the cubic center is 6 using the fifth order averaging method. Moreover, both of the numbers can be reached by analyzing the number of simple zeros of the obtained averaged functions. In some sense, our results generalize the results in [Appl.

Math. Comput. 250(2015), 887–907], Theorems 1 and 2 to the piecewise systems class.

Keywords: averaging method, center, piecewise differential systems, limit cycle, peri- odic orbits.

2010 Mathematics Subject Classification: 34C05, 34C07.

1 Introduction and main results

One of the main open problems in the qualitative theory of real planar differential systems is the determination and distribution of limit cycles. There are several methods for studying the bifurcation of limit cycles. One of the methods is by perturbing a differential system which has a center. In this case the perturbed system displays limit cycles that bifurcate, either from some of the periodic orbits surrounding the center, or from the center (having the so-called Hopf bifurcation), see the book of Christopher–Li [4], and references cited therein.

The problem of bounding the number of limit cycles for planar smooth differential systems has been exhaustively studied in the last century and is closed related to the 16th Hilbert’s problem [10,13]. Solving this problem even for the quadratic case seems to be out of reach at the present state of knowledge. In the last few years there has been an increasing interest in

BEmail: huangbo0407@126.com

the study of discontinuous piecewise differential systems, see [3,7,11,14,18,21] for instance.

This interest has been mainly motivated by their wider range of application in various fields of science (e.g., control theory, biology, chemistry, engineering, physics, etc.).

Our goal in this paper is to study the bifurcation of limit cycles for a class of cubic poly- nomial differential systems having a rational first integral of degree 2. We remark that the classification of all cubic polynomial differential systems having a center at the origin and a rational first integral of degree 2 can be found in [17]. Later on, the authors in [16] summa- rized this classification in six families of cubic polynomial differential systems. In particular they obtained the class

˙

x =2y(x+α)², y˙ =−2(x+α)(αx−y²), (1.1) whereα6=0. System (1.1) called classP6in [16], which hasH(x,y) = ₍^x2+y2

α+x)² as its first integral with the integrating factor µ(x,y) = 1/(α+x)⁴. See [16] for the phase portraits of system (1.1) in the Poincaré disk.

A natural question is: What happens with the periodic orbits (or the center) of the system(1.1) when it is perturbed inside the class of all smooth cubic polynomial differential systems, or inside the class of all discontinuous piecewise cubic polynomial differential systems with a straight line of discontinuity?

In this article we say a medium limit cycle is one which bifurcates from a periodic orbit surrounding a center, and asmall limit cycleis one which bifurcates from a center equilibrium point. Remark that, for the piecewise cubic polynomial vector fields there are two recent works, see [8,9], obtaining at least 18 and 24 small limit cycles, respectively. Our objective in this paper is to study the maximal number of medium and small limit cycles for the cubic center (1.1), when they are perturbed inside the class of all discontinuous piecewise cubic polynomial differential systems with the straight line of discontinuityy =0. The main results are based on the averaging method. We remark that the method of averaging is a classic and mature tool for studying the behaviour of nonlinear differential systems in the presence of a small parameter. For more details about this method see the book of Sanders, Verhulst and Murdock [24] and Llibre, Moeckel and Simó [19].

More precisely, we consider the following discontinuous piecewise polynomial differential systems

x˙

˙ y

=

2y(x+α)²

−2(x+α)(αx−y²)

+ε











p₁(x,y) q1(_x,y)

, y>0, p₂(x,y)

q₂(x,y)

, y<0,

(1.2)

where

p₁(x,y) =

∑

0≤i+j≤3

a_i,jxⁱy^j, q₁(x,y) =

∑

0≤i+j≤3

b_i,jxⁱy^j, p₂(x,y) =

∑

0≤i+j≤3

c_i,jxⁱy^j, q₂(x,y) =

∑

0≤i+j≤3

d_i,jxⁱy^j. (1.3) Moveover, we consider the following smooth polynomial differential systems















˙

x =2y(x+α)²+

∑

ε^sµ_s(x,y),

˙

y =−2(x+α)(αx−y²) +

∑

ε^sνs(x,y),

(1.4)

and the discontinuous piecewise cubic polynomial differential systems x˙

˙ y

=

2y(x+α)²

−2(x+α)(αx−y²)

+

∑

ε^s











µs(x,y) ν_s(x,y)

, y>0, ψ_s(x,y)

φs(x,y)

, y<0,

(1.5)

where

µ_s(x,y) =

∑

0≤i+j≤3

a_s,i,jxⁱy^j, ν_s(x,y) =

∑

0≤i+j≤3

b_s,i,jxⁱy^j, ψ_s(x,y) =

∑

0≤i+j≤3

c_s,i,jxⁱy^j, φ_s(x,y) =

∑

0≤i+j≤3

d_s,i,jxⁱy^j. The main results of this paper are stated as follows.

Theorem 1.1. For |ε| > 0 sufficiently small the maximum number of medium limit cycles of the discontinuous piecewise differential system(1.2) is 9 using the first order averaging method, and this number can be reached.

Ifa_i,j =c_i,jandb_i,j = d_i,j (see (1.3)), then the perturbed system (1.2) is smooth. In this case, we obtain the following corollary of Theorem1.1.

Corollary 1.2. When a_i,j =c_i,j and b_i,j =d_i,j, the maximum number of medium limit cycles of system (1.2)that bifurcate using the first order averaging method is 3 and it is reached.

Remark 1.3. Theorem1.1gives the exact upper bound of the number of limit cycles bifurcated from the periodic orbits of the cubic center (1.1), which is challenging. Theorem 1.1 and Corollary1.2 show that the maximum number of limit cycles for the piecewise case is 6 more than the smooth one. We note that the smooth case of system (1.2) has been studied in [16, Section 3.3] under the condition a_0,0 = b_0,0 = c_0,0 = d_0,0 = 0. Corollary 1.2 shows that the non-zero constant terms provide no more information on the limit cycles. However, in the piecewise case, with the non-zero constant terms the perturbed system (1.2) can produce at least one more limit cycle than the case without them (see Remark 3.1 in Section3). This phenomenon coincides with the well-known pseudo-Hopf bifurcation (see [2,6]).

Theorem 1.4. For|ε|>0sufficiently small using the fifth order averaging method, we obtain that (a) for any real constants a_s,i,j and b_s,i,j (s = 1, . . . , 5, 0 ≤ i+j ≤ 3) with a_1,0,0 = b_1,0,0 = 0,

system (1.4)has at most 2 small limit cycles bifurcating from the center(1.1), and this number can be reached;

(b) system(1.5)has at most 6 small limit cycles bifurcating from the center(1.1)under the condition a_1,0,0 =b_1,0,0 =c_1,0,0=d_1,0,0=0, and this number can be reached.

More concretely, we provide in Table1.1the maximum number of limit cycles for systems (1.4) and (1.5) up to thei-th order averaging method fori=1, . . . , 5.

The outline of this paper is as follows. In Section 2, we introduce the basic theory of the averaging method for discontinuous piecewise planar differential systems. The averaged function associated to system (1.2) is obtained in Section3. Section 4focuses on the analysis of the exact upper bound for the number of zeros of the averaged function, and the theory of Chebyshev systems is used to prove Theorem 1.1. The objective of Section 5 is to study the small limit cycles of systems (1.4) and (1.5). Finally, we present the explicit formulae of the i-th order averaged function up toi=5 in AppendixAfor reference.

Averaging order System (1.4) System (1.5)

1 0 1

2 0 2

3 1 4

4 1 6

5 2 6

Table 1.1: Number of small limit cycles for systems (1.4) and (1.5).

2 Preliminary results

In this section we introduce the basic theory of the averaging method that we shall use in our study of the cubic center (1.1). The following result is due to Itikawa, Llibre and Novaes [14].

Consider the discontinuous piecewise differential systems of the form dr

dθ =r⁰ =

(F⁺(θ,r,ε), if 0≤θ ≤γ,

F⁻(θ,r,ε), ifγ≤ θ≤2π, (2.1) where

F^±(θ,r,ε) =

∑

εⁱF_i^±(θ,r) +ε^k+1R^±(θ,r,ε),

andεis a real small parameter. The set of discontinuity of system (2.1) is∑={θ =0} ∪ {θ= γ}_{if 0} < γ < _{2π. Here} F_i^± : S¹×D →_{R for} i= _{1, . . . ,}k, and R^± : S¹×D×(−ε₀,ε₀) → _R are C^k functions, being D an open and bounded interval of (0,∞), ε₀ is a small parameter, andS¹≡_R/(2π). This last convention implies that the functions involved in system (2.1) are 2π-periodic in the variableθ.

The averaging method consists in defining a collection of functions f_i : D→R, called the i-th order averaged function, fori =1, 2, . . . ,k, which control (their simple zeros control), for

|ε| > 0 sufficiently small, the isolated periodic solutions of the differential system (2.1). In Itikawa–Llibre–Novaes [14] it has been established that

f_i(z) = ^y

+

i (γ,z)−y_i⁻(γ−2π,z)

i! , (2.2)

where y^±_i : S¹×D → _{R, for}i = 1, 2, . . . ,k, are defined recurrently by the following integral equations

y^±₁(θ,z) =

Z _θ

0 F₁^±(ϕ,z)dϕ, y^±_i (θ,z) =i!

Z _θ

0

F_i^±(ϕ,z) +

∑

`=1

∑

S`

1

b₁!b₂!2!^b2· · ·b_{`</sub>!`_!<sup>b`</sup> ·∂<sup>L</sup>F<sub>i</sub><sup>±</sup><sub>−`}(ϕ,z)

∏

y^±_j (ϕ,z)^bjdϕ, (2.3)

whereS<sub>`</sub> is the set of all`-tuples of non-negative integers[b₁,b₂, . . . ,b_`]_satisfying b₁+2b₂+

· · ·+`b` =`andL=b<sub>1</sub>+b<sub>2</sub>+· · ·+b`. Here,∂^LF(ϕ,z)denotes the Fréchet’s derivative with respect to the variablez. We remark that, the investigation in this paper is restricted to F₀ =0 in expression (2.3). For the general form of the averaged functions see [20].

We point out that takingγ= 2π system (2.1) becomes smooth. So the averaging method described above can also apply to smooth differential systems. In practical terms, the evalu- ation of the recurrence (2.3) is a computational problem that requires powerful computerized resources. Usually, the higher the averaging order is, the more complex are the computational operations to calculate the averaged function (2.2). Recently in [22] the Bell polynomials were used to provide a relatively simple alternative formula for the recurrence (2.3). And based on this new formula, an algorithmic approach to revisit the averaging method was introduced in [12] for the analysis of bifurcation of small limit cycles of planar differential systems. More- over, we provide an upper bound of the number of zeros of the averaged functions for the general class of perturbed differential systems (see Theorem 3.1 in [12]).

The following k-th order averaging theorem gives a criterion for the existence of limit cycles. Its proof can be found in Section 2 of [14].

Theorem 2.1 ([14]). Assume that, for some j ∈ {1, 2, . . . ,k}, f_i = 0 for i = 1, 2, . . . ,j−1 and f_j 6=0. If there exists z^∗ ∈ D such that f_j(z^∗) 6= 0, then for|ε|> ₀sufficiently small, there exists a 2π-periodic solution r(θ,ε)of (2.1)such that r(0,ε)→z^∗whenε →0.

The following theorem (see Theorem 5.2 of [1] for a proof) provides an approach to trans- form a perturbed differential system into the standard form (2.1), which can be used to apply the first order averaging method.

Theorem 2.2([1]). Consider the differential system

˙

x=P(x,y) +εp(x,y),

˙

y=Q(x,y) +εq(x,y), (2.4)

where P,Q,p and q are continuous functions in the variables x and y, and ε is a small parameter.

Suppose that system (2.4)_ε=0 has a continuous family of ovals

Γh ⊂ (x,y)|H(x,y) = h,h ∈ (h₁,h2) , where H(x,y)is a first integral of(2.4)_ε=0, and h₁ and h2 correspond to the center and the separatrix polycycle, respectively. For a given first integral H = H(x,y), assume that xQ(x,y)− yP(x,y) 6= 0 for all(x,y) in the periodic annulus formed by the ovals{_Γh}. Let ρ : (√

h₁,√ h₂)× [0, 2π)→[0,+_∞)be a continuous function such that

H(ρ(R,ϕ)cosϕ,ρ(R,ϕ)sinϕ) =R², for all R ∈ (√

h₁,√

h₂) and all ϕ ∈ [0, 2π). Then the differential equation which describes the dependence between the square root of energy R= √

h and the angle ϕfor system(2.4)is dR

dϕ = εµ(x²+y²)(Qp−Pq)

2R(Qx−Py) +O(ε²), (2.5)

where µ = µ(x,y) is the integrating factor of system(2.4)_ε=0 corresponding to the first integral H, and x=ρ(R,ϕ)cosϕand y =ρ(R,ϕ)sinϕ.

In general, it is not an easy thing to deal with zeros of the averaged function (2.2). The techniques and arguments to tackle this kind of problem are usually very long and technical.

In what follows we present some effective results on obtaining the lower bound and the upper bound of the number of zeros for a complicated function. The next result is used to obtain a lower bound of the number of simple zeros for an averaged function.

Lemma 2.3 ([5]). Consider n+1 linearly independent analytical functions f_i(x) : A → _R,i = 0, 1, . . . ,n, where A ⊂ _R is an interval. Suppose that there exists k ∈ {0, 1, . . . ,n} such that f_k(x) has constant sign. Then there exist n+1constants c_i,i = 0, 1, . . . ,n such that c₀f₀(x) +c₁f₁(x) +

· · ·+c_nf_n(x)has at least n simple zeros in A.

It is important to point out that the classical theory of Chebyshev systems is useful to provide an upper bound for the number of zeros. Let F = [f₀, . . . ,f_n] be an ordered set of functions of class Cⁿ defined in the closed interval [a,b]. We consider only elements in Span(F), that is, functions such as f = a0f0+a₁f₁+· · ·+anfn where a_i, fori = 0, 1, . . . ,n, are real numbers. When the maximum number of zeros, taking into account its multiplicity, isn, the set F is called an Extended Chebyshev system (ET-system) in[a,b]. We say that F is an Extended Complete Chebyshev system (ECT-system) in [a,b], if any set [f0, f₁, . . . ,f_k], for k = 0, . . . ,n is an ET-system. When all the Wronskians, W_k := W(f₀, f₁, . . . ,f_k) 6= 0 for 0 ≤ k ≤ n in [a,b] the set F is an ECT-system. For more details on ET-systems and ECT- system, see [15] for instance.

We remark that not always the standard study of ET-systems can be applied to bound the number of zeros of elements in Span(F). Here we use an extension of this theory (see [23]). The following result provides an effective estimation for the number of isolated zeros of elements in Span(F)when some Wronskians vanish.

Theorem 2.4([23]). LetF = [f₀,f₁, . . . ,f_n]be an ordered set of analytic functions in[a,b]. Assume that all theν_i zeros of the Wronskian W_i are simple for i = 0, 1, . . . ,n. Then the number of isolated zeros for every element of Span(F)does not exceed

n+νn+ν_n−1+2(νn−2+· · ·+ν0) +λ_n−1+· · ·+λ3, whereλ_i =min(2νi,ν_i−3+· · ·+ν₀), for i=_{3, . . . ,}n−1.

3 Averaged function associated to system (1.2)

In this section we will get the first order averaged function associated to system (1.2) by using Theorem 2.1. We remark that the period annulus of the differential system (1.1) is formed by the ovals {_Γh} ⊂ {(x,y)|H(x,y) = h,h ∈ (0, 1)}. By solving implicitly the equation H(ρ(R,ϕ)cosϕ,ρ(R,ϕ)sinϕ) =R²we obtain the positive functionρ(R,ϕ)given by

ρ(R,ϕ) =−^αR(signum(α) +Rcosϕ) R²cos²ϕ−1

forϕ∈[0, 2π)andR∈(0, 1), where signum(α)is the sign function defined by signum(α) =

( 1, α>0,

−1, α<0.

Using Theorem2.2, we can transform system (1.2) into the form dR

dϕ =







ε⁻⁽_4αR^Qp₍¹_x⁻₊^Pq_α)¹5⁾

x=ρ(R,ϕ)cosϕ,y=ρ(R,ϕ)sinϕ

+O(ε²), 0≤ ϕ≤π, ε⁻⁽_4αR^Qp₍²_x⁻₊^Pq2)

α)⁵

x=ρ(R,ϕ)cosϕ,y=ρ(R,ϕ)sinϕ

+O(ε²)_, π≤ ϕ≤2π. (3.1)

Now the discontinuous piecewise differential system (3.1) is under the assumptions of Theo- rem2.1. Next, we will study the zeros of the averaged function f :(0, 1)→_Rgiven by

f(R) =

Z _π

0

−(Qp₁−Pq₁) 4αR(x+α)⁵

x=ρ(R,ϕ)cosϕ,y=ρ(R,ϕ)sinϕ

dϕ +

Z _2π

π

−(Qp₂−Pq₂) 4αR(x+α)⁵

x=ρ(R,ϕ)cosϕ,y=ρ(R,ϕ)cosϕ

dϕ

=

Z _π

0

A(ϕ;a,b)cosϕ+B(ϕ;a,b) 2α³(signum(α)·Rcosϕ−1)^dϕ+

Z _2π

π

A(ϕ;c,d)cosϕ+B(ϕ;c,d) 2α³(signum(α)·Rcosϕ−1)^dϕ, where

A(_ϕ;a,b) = −R³

α³(_a0,3−a2,1−b3,0+_b1,2) +α²(−b0,2+_b2,0+_a1,1) +α(−b_1,0−a_0,1) +b_0,0

S³+signum(α)·R²

α³R²(a_1,2−a_3,0) +α²(R²(a_2,0−a_0,2)−a_0,2+a_2,0−b_1,1) +α(−R²a_1,0−2a_1,0+2b_0,1) + (R²+3)a_0,0

S²−R

α³R²(a_2,1+b_3,0)−α²R²(2a_1,1+b_2,0) +α(R²(3a_0,1+b_1,0) +a_0,1+b_1,0)−(R²+3)b_0,0

S+signum(α)

·α³R⁴a_3,0−α²R²(R²+1)a_2,0+αR²(R²+3)a_1,0−(R⁴+6R²+1)a_0,0 , B(ϕ;a,b) =R³[α³(−b_0,3+b_2,1+a_1,2−a_3,0) +α²(−b_1,1+a_2,0−a_0,2)

+α(−a_1,0+b_0,1) +a0,0]S⁴+signum(α)·R²[α³R²(a0,3−a_2,1) +α²((R²+₁)a_1,1−b_0,2+b_2,0)−α((R²+₂)a_0,1+2b_1,0) +3b_0,0]S³

−R[α³R²(a_1,2−2a_3,0+b_2,1) +α²R²(−2a_0,2+3a_2,0−b_1,1) +α(R²(−4a_1,0+b_0,1)−a_1,0+b_0,1) + (5R²+3)a_0,0]S² +signum(α)·[α³R⁴a_2,1−α²R²((R²+1)a_1,1+b2,0) +αR²((R²+3)a_0,1+2b_1,0)−(3R²+1)b_0,0]S

−R[α³R²a_3,0−2α²R²a_2,0+α(3R²+1)a_1,0−4(R²+1)a_0,0]

with S = sinϕ, and a = (a_i,j), b = (b_i,j), c = (c_i,j)andd = (d_i,j), with a_i,j,b_i,j,c_i,j andd_i,j are parameters appearing in the perturbed polynomials (1.3).

Computing the above integrals and making the transformation R = ^2ω

1+ω² for 0< ω < 1 we obtain

f(_R)^R

= ^2ω

1+ω2

= ^f˜(ω)

6α³ω(ω²+1)³ = ^∑

8i=0k_if_i(ω)

6α³ω(ω²+1)^{3, (3.2)} where

f₀(ω) =ω², f₁(ω) =ω⁴, f₂(ω) =ω⁶,

f₃(ω) =ω⁸, f₄(ω) =ω⁵+ω³, f₅(ω) =ω⁷+ω, f₆(ω) =ω⁴ln

1+ω 1−ω

, f₇(ω) = (ω⁸+1)ln

1+ω 1−ω

, f₈(ω) = (ω⁶+ω²)ln

1+ω 1−ω

,

(3.3)

and

k₀ =−_3π−α(a_1,0+c_1,0)−α(b_0,1+d_0,1) +₄(a_0,0+c_0,0)_, k₁ =−3π

−3α³(a_3,0+c_3,0)−3α³(b_0,3+d_0,3)−α³(a_1,2+c_1,2)

−α³(b_2,1+d_2,1) +4α²(a_0,2+c_0,2) +4α²(a_2,0+c_2,0)−6α(a_1,0+c_1,0)

−2α(b_0,1+d_0,1) +12(a_0,0+c_0,0), k₂ =−3π

2α³(a_1,2+c_1,2) +2α³(a_3,0+c_3,0)−2α³(b_0,3+d_0,3)−2α³(b_2,1+d_2,1)

−α(a_1,0+c_1,0)−α(b_0,1+d_0,1) +₄(a_0,0+c_0,0)_, k₃=3πα³

(a_1,2+c_1,2)−(a_3,0+c_3,0)−(b_0,3+d_0,3) + (b_2,1+d_2,1), k₄=signum(α)·^h−2α³(a_2,1−c_2,1)−22α³(a_0,3−c_0,3) +2α³(b_1,2−d_1,2)

−26α³(b_3,0−d_3,0) +8α²(b_2,0−d_2,0) +8α²(a_1,1−c_1,1) +16α²(b_0,2−d_0,2)

−32α(a_0,1−c_0,1)−8α(b_1,0−d_1,0) +26(b_0,0−d_0,0)ⁱ,

k5=signum(α)·^h6α³(a0,3−c0,3) +6α³(b_1,2−d_1,2)−6α³(b3,0−d3,0)

−6α³(a_2,1−c_2,1) +6(b_0,0−d_0,0)ⁱ, k₆ =−signum(α)·6α³

3(a_0,3−c_0,3) + (a_2,1−c_2,1)−(b_1,2−d_1,2)−3(b_3,0−d_3,0), k₇ =−signum(α)·3α³

(a_0,3−c_0,3)−(a_2,1−c_2,1) + (b_1,2−d_1,2)−(b_3,0−d_3,0), k₈=_signum(α)·_12α³(a_0,3−c_0,3) + (b_3,0−d_3,0)_.

It follows directly from

∂(k₀,k₁,k₂,k₃,k₄,k₅,k₆,k₇,k₈)

∂(b0,0,a3,0,a_1,2,a_1,0,a_1,1,a2,0,b3,0,a_2,1,a0,3) =_signum(α)·_107495424π⁴α²⁰6=₀

that the constantsk₀,k₁,· · · ,k₈are independent. That is to say, the coefficients of the functions f_i(ω), i = 0, 1, . . . , 8 can be chosen arbitrarily. Moreover, the functions f₀(ω), . . . ,f₈(ω) are linearly independent. In fact, we obtain the following Taylor expansions in the variable ω aroundω =0 for these functions:

f₀(ω) =ω², f₁(ω) =ω⁴, f₂(ω) =ω⁶,

f₃(ω) =ω⁸, f₄(ω) =ω⁵+ω³, f₅(ω) =ω⁷+ω, f₆(ω) =_2ω⁵+ ²

3ω⁷+²

5ω⁹+O(ω¹¹)_, f₇(ω) =2ω+²

3ω³+²

5ω⁵+ ²

7ω⁷+²⁰

9 ω⁹+O(ω¹¹), f₈(ω) =2ω³+ ²

3ω⁵+¹²

5 ω⁷+ ²⁰

21ω⁹+O(ω¹¹).

(3.4)

The determinant of the coefficient matrix of the variables ω,ω², . . . ,ω⁹ is equal to 8388608/496125. Hence, by Lemma 2.3 it follows that there exists a linear combination of f_i(ω), i=0, 1, . . . , 8 with at least 8 simple zeros, which means that system (1.2) has at least 8 limit cycles bifurcating from the period orbits surrounding the origin.

Remark 3.1. We notice that when the constant terms a_0,0, b_0,0, c_0,0, d_0,0 are identically zeros.

In a similar way, we can prove that system (1.2) has at least 7 limit cycles bifurcating from the period orbits surrounding the origin. In fact, k₅+2k₇ = 0 in this case, and the function ˜f(ω) in (3.2) is a linear combination of 8 linearly independent functions f₀, . . . ,f₄,f₆,f₇−2f₅, f₈. Therefore, by Lemma 2.3, the perturbed system (1.2) with the non-zero constant terms can produce at least one more limit cycle than the case without them.

Proof of Corollary1.2. Obviously, when a_i,j = c_i,j andb_i,j = d_i,j, the coefficients k₄,k₅, . . . ,k₈ are identically zeros. It is easy to check that (f₀, . . . ,f₃) is an ECT-system. Therefore, the averaged function f in this case has at most 3 simple zeros and this number can be reached.

Hence, by Theorem2.1, Corollary1.2is proved.

In what follows, we first provide an upper bound of the number of zeros of the function f˜(ω) in (3.2). We eliminate the logarithmic function by taking the ninth derivative of ˜f(ω) and obtain

f˜⁽⁹⁾(ω) =_signum(α)· ^110592α3

(1+ω)⁹(−1+ω)⁹(H₁ω⁸+H₂ω⁶+H₃ω⁴+H₂ω²+H₁)_, where

H₁ =−14(a_2,1−c_2,1) +14(b_1,2−d_1,2) +8(a0,3−c0,3)−83(b3,0−d3,0), H2 =−24(_a2,1−c2,1) +₂₄(_b1,2−d1,2)−32(_a0,3−c0,3)−1988(_b3,0−d3,0)_, H₃ =76(a_2,1−c_2,1)−76(b_1,2−d_1,2) +48(a_0,3−c_0,3)−4818(b_3,0−d_3,0).

As a result of the symmetry of coefficients of the function ˜f⁽⁹⁾(ω)with respect toω, it is easy to know that the zeros of the function ˜f⁽⁹⁾(ω) appear in pairs. Recalling this property, we obtain that ˜f⁽⁹⁾(ω) has at most 2 zeros in (0, 1). Thus, by using Rolle’s theorem and noting the fact that ˜f(0) = 0, we conclude that ˜f(ω) has at most 2+9−1 = 10 zeros in (0, 1), which means that system (1.2) has at most 10 limit cycles bifurcating from the period orbits surrounding the origin. In next section, we will show that the bound of the number of limit cycles can be reduced to 9 by Theorem2.4. Moreover, this number can be reached.

4 Proof of Theorem 1.1

In this section we will study the maximum number of simple zeros of the averaged function (3.2). The main effort is based largely on algebraic calculations with the theory of Chebyshev systems used to deal with the Wronskian determinants.

First, we denote byW_i(ω)the Wronskian for the functions f₀, f₁, . . . ,f_i depending onω:

W_i(ω) =W(f₀, . . . ,f_i), i=0, 1, . . . , 8.

Next, we will show that all the Wronskians have no zeros exceptW₇(ω) which vanishes at a unique zero in (_{0, 1}), which is simple. Using the expressions in (3.3), we perform the

calculation and obtain

W₀(ω) =ω², W₁(ω) =2ω⁵, W₂(ω) =16ω⁹, W₃(ω) =768ω¹⁴, W₄(ω) =2304ω¹³(3ω²−5), W₅(ω) =69120ω⁹(1−ω²)(3ω⁶−7ω⁴−7ω²+35), W₆(ω) =−^3317760ω

8(ω²+1) (1−ω²)^{5 T6}(ω), W7(ω) =−133772083200ω(ω²+₁)³T_7,0(ω)

(1−ω²)⁴

ln

1+ω 1−ω

− ^2ωT7,1(ω) 105(1−ω²)⁶T_7,0(ω)

, W₈(ω) = 821895679180800(ω²+1)⁶

(1−ω²)¹⁰

T_8,0(ω)·ln

1+ω 1−ω

+ ^2ωT8,1(ω) 105(1−ω²)⁴

,

(4.1)

where

T₆(ω) =15ω¹⁴−195ω¹²−89ω¹⁰+1149ω⁸+421ω⁶−4305ω⁴+805ω²−105<0, T_7,0(ω) =15ω⁸−140ω⁶+1018ω⁴−140ω²+15>0,

T_7,1(ω) =160ω²⁰−8569ω¹⁸+105687ω¹⁶−547324ω¹⁴+1437092ω¹²−2101414ω¹⁰ +1752730ω⁸−839580ω⁶+210980ω⁴−23625ω²+1575,

T_8,0(ω) =35ω⁸−1100ω⁶+2898ω⁴−1100ω²+35,

T_8,1(ω) =45477ω¹⁴−444465ω¹²+1433397ω¹⁰−2210985ω⁸+1803095ω⁶

−745675ω⁴+128975ω²−3675,

(4.2)

by Sturm’s theorem. It is easy to judge thatW_i(ω)fori=0, . . . , 6 does not vanish in the open interval(0, 1). The difficulties mainly focus on the determination ofW₇(ω)andW₈(ω). Proposition 4.1. W₇(ω)has a unique zero inω∈ (0, 1)and this zero is simple.

Proof. Denote the function in the parentheses ofW₇(ω)byQ₇(ω), then Q₇⁰(ω) = ^64ω

6(ω²+1)(5ω⁸+172ω⁶−1122ω⁴+172ω²+5)T₆(ω) 105(1−ω²)⁷T_7,0² (ω)

has a unique simple zeroω^∗ in(0, 1)and can be easily isolated (e.g. by using the command realroot(%, 1/10000) in Maple) as ω^∗ ∈ ¹¹²⁰⁸⁷₂₆₂₁₄₄,¹⁴⁰¹¹₃₂₇₆₈

. It follows that Q₇(ω) decreases in (0,ω^∗)and increases in(ω^∗, 1). Note also that lim

ω→0⁺Q7(ω) =0 and lim

ω→1⁻Q7(ω) = +_{∞. Thus,} Q₇(ω)has a unique simple zero in(_{0, 1}), equivalently,W₇(ω)has a simple zero in(_{0, 1})_{. This} ends the proof.

Proposition 4.2. W₈(ω)does not vanish inω ∈(0, 1).

Proof. First, using Sturm’s theorem, we get that T_8,0(ω) has two simple zeros ω₁ and ω₂ in (0, 1) and T_8,1(ω) has three simple zeros ω₃, ω₄ and ω₅ in (0, 1), and these zeros can be respectively isolated as

0.18709157≈ω₁∈ _33554432^6277751,_4194304⁷⁸⁴⁷¹⁹ , 0.64417845≈ω2∈ ³³⁷⁷³⁵₅₂₄₂₈₈,^5403761_8388608

, 0.18709131≈ω₃∈ _16777216^3138871_,33554432^6277743_, 0.66278355≈ω₄∈ ^5559831_8388608,_1048576⁶⁹⁴⁹⁷⁹

, 0.75595958≈ω₅∈ _1048576⁷⁹²⁶⁸¹,^6341449_8388608

.

We denote the function in the parenthesis ofW₈(ω)byQ₈(ω), it is easy to verify thatQ₈(ω₁)6=

0 and Q8(ω₂) 6= 0. In order to study the number of zeros of Q8(ω) in (0, 1) we define a functionZ₈(ω)as follows

Z8(ω):=^Q8(ω) T_8,0(ω) =ln

1+ω 1−ω

+ ^2ωT8,1(ω)

105(1−ω²)⁴T_8,0(ω)^{, ω} ∈(0, 1)\{ω₁,ω2}. It is obvious that the function Z₈(ω)has the following properties (see Fig.4.1):

lim

ω→ω⁻₁

Z₈(ω) = +_∞, lim

ω→ω₁⁺

Z₈(ω) =−_∞, lim

ω→ω⁻₂

Z₈(ω) =−_∞, lim

ω→ω₂⁺

Z₈(ω) = +_∞.

A direct calculation shows that

Z₈⁰(ω) = ^32768ω

8(ω²+1)H₈(ω) 35(1−ω²)⁵T_8,0² (ω) ^, where

H₈(ω) =35ω¹⁴+85ω¹²−129ω¹⁰−503ω⁸−119ω⁶+1855ω⁴−875ω²+35.

Obviously, H₈(ω)has two simple zerosω₁^∗andω^∗₂ in(0, 1)and can be respectively isolated as 0.21002672≈ω₁^∗ ∈_2147483648^451028943,_4294967296^902057887

, 0.69221454≈ω₂^∗ ∈^185814925_268435456,_1073741824^743259701

. (4.3)

Therefore Z8(ω) increases when ω ∈ (0,ω₁)∪(ω₁,ω₁^∗) and ω ∈ (ω^∗₂, 1); decreases when ω∈(ω^∗₁,ω₂)∪(ω₂,ω₂^∗)(see Fig.4.1). Notice that

lim

ω→0⁺Z₈(ω) =0, lim

ω→1⁻Z₈(ω) = +_∞.

It follows from (4.3) that

Z₈(ω^∗₁)≈ −0.0000126678<0, Z₈(ω₂^∗)≈1.126483743>0.

Taking into account the above results, we conclude that Z₈(ω) does not vanish for ω ∈ (0, 1)\{ω₁,ω₂}. Thus the desired result follows.

Proof of Theorem1.1. It follows from equation (4.1), Propositions 4.1 and 4.2 that W_i(ω), i = 0, 1, . . . 6 andW8(ω)do not vanish in the interval (_{0, 1})_{, and W7}(ω)has exactly 1 simple zero in (0, 1). Thus F = [f₀,f₁, . . . ,f₈] defined in (3.3) satisfies the assumptions of Theorem 2.4, which implies that any linear combination of f₀,f₁, . . . ,f₈ can possess at most 9 zeros in (0, 1), counting with multiplicities. But the authors in [23] do not prove that the upper bound can be reached in the general cases. In what follows we will show that the upper bound 9 can be reached in our system.

Following the ideas of [23], we first look for an element in Span(F) with a zero of the highest multiplicity, then we perturb it inside Span(F)in order to have the prescribed con- figuration of zeros. We remark that since the Wronskian determinantW₈(ω)does not vanish,

Figure 4.1: The curveZ₈(ω)does not vanish forω ∈(0, 1)\{ω₁,ω₂}.

(a) (b)

Figure 4.2: Two cases for G(ω) having 9 zeros in (0, 1) taking into account multiplicity. In particularω0has multiplicity 8.

the averaged function (an element in Span(F)) can not have a zero in (0, 1) with multiplic- ity 9. Then we try to find an elementG(ω) =_∑⁷_i=0a_if_i+k f₈∈ _Span(F), of which has a zero ω₀ ∈ (0, 1)with multiplicity 8. Note that G(ω) has 9 zeros in (0, 1) with ω₀ of multiplicity 8 may have two cases as shown in Fig. 4.2. For the generation of such ω₀ we provide an algorithm (Maple program) in AppendixB.

Now letω₀=781/10001,K₀ =ln

1+ω₀ 1−ω0

andk =10⁸. Consider the function

G(ω) =_a0f0(ω) +_a1f1(ω) +· · ·+_a7f7(ω) +_{k f8}(ω)_, ω∈(_{0, 1})_. (4.4) By direct calculation we get the power series ofGaround the pointω₀:

G(ω) =e₀+e₁(ω−ω₀) +· · ·+e₇(ω−ω₀)⁷+e₈(ω−ω₀)⁸+· · · , wheree_i is the linear combination of a₀,a₁, . . . ,a₇. We solve the equations

e₀ =_0, e₁=_{0, . . . ,} e₇ =_0,

and find the values ofa₀,a₁, . . . ,a₇which have the form a_i = ^∑

ji

j=0L_i,jK₀^j

k₁K₀+k₂ , i=0, . . . , 7, (4.5) where

k₁=585397408871072540089139375831993705697245971 45302237853421492432853598362240000000,

k₂ =−916164764498521481287490087182092157549 2097096776449170037730387965998807150160399, and

j_i =







2, i∈ {0, 1, 2}, 1, i∈ {3, 4, 5, 6}, 0, i∈ {7},

and each L_i,j in (4.5) is an integer or rational with high number of digits in numerators and denominators. We will not write down here the explicit expression ofa_i for the sake of brevity.

It turns out that

G(ω) =e₈(ω−ω₀)⁸+O((ω−ω₀)⁹)_, ω→ω₀, (4.6) where

e8 =− ^k3·(B₁K₀+B₀)

625678681207969855947716482401·(k₁K₀+k₂)^, with

k₃ =6373960409705365063968756422951747001176840429758709070500, B₁ =2371833114839857298494412882156005750986234376264757348752800000, B₀ =−371199090602328323784582373340236998424005450432934748931637759, ande8 ≈6.468110730×10⁷. On the other hand, the Taylor expansion ofG(ω)nearω =0 is

G(ω) =C₁ω+O(ω²), (4.7)

where

C₁ = ^k4· k₅K₀−k₆

55588252797009·(k₁K₀+k₂) ≈ −3.242325599 with

k₄=227096370975140733661254232304854673313104068100000, k₅=864359913055284073500033389565682256669487378000, k₆=135274953622915880496646897785052547295533923181.

By the way, we would like to point out that our purpose of choosing such ak in (4.4) is to make the expressions of the numberse₈andC₁to be relative simple. Equations (4.6) and (4.7) mean that (i) Ghas a zero at ω₀ with multiplicity 8, (ii) there exists an ε₀ with 0 < ε₀ < ω₀ such that G(ω)is positive in [ε₀,ω₀), and (iii)G(ω)is negative nearω =0. Moreover, G(ω)

is positive in(ω₀, 1)because lim

ω→1⁻G(ω) = +_∞ (otherwiseG(ω)would has 10 zeros in(0, 1) counting multiplicity, which leads to a contradiction).

Fixing the numbersa₀,a₁, . . . ,a₇ andk, we consider the function G_ε(ω) =G(ω) +

∑

ε_if_i(ω), ω∈ (0, 1). (4.8) We note that f_i can be extended analytically to [0, 1). Thus there exists a small number M>0 such that

Gε(ε₀)> ¹

2G(ε₀)>0, Gε(ω)< ¹

2C₁ω <0, whenω→0⁺, lim

ω→1⁻Gε(ω) = +_∞,

for all|ε_i|< M,i=0, 1, . . . , 8. Moreover, nearω₀ we find

∑

ε_if_i(ω) =µ₀+µ₁(ω−ω₀) +· · ·+µ₈(ω−ω₀)⁸+· · · ,

whereµ_i =µ_i(ε₀,ε₁, . . . ,ε₈)is linear combination ofε₀,ε₁, . . .,ε₈. One can directly check that the matrix of the coefficients ofµ₀,µ₁, . . . ,µ₈with respect toε₀,ε₁, . . . ,ε₈has rank 9, and hence µ₀,µ₁, . . . ,µ₈ are independent.

Consequently, since f_i is analytic at ω0 and G(ω)has a zero at ω0 with multiplicity 8, it follows that there exists some small|ε_i| M (i=0, 1, . . . 8)(and henceµ_i is small) such that G_ε has exactly 8 simple zeros in a small enough neighborhood of ω₀. In view of (4.8) G(ω) has an extra zero in(0,ε0). According to the result of [23], this zero is simple. That is to say, Gε has 9 simple zeros.

Finally, taking into account the above analysis, we see that system (1.2), up to the first order averaging method, has at most 9 limit cycles, and the upper bound can be reached. The proof of Theorem1.1is finished.

Remark 4.3. If ¯Ris a simple zero of the averaged function f(R)(see (3.2)), by Theorem2.1we have a limit cycleR(ϕ,ε)of the differential system (3.1) such thatR(0,ε) =R^¯+O(ε). Then, go- ing back through the changes of variables (see (3.1)) we have for the discontinuous piecewise differential system (1.2) the medium limit cycle (x(t,ε),y(t,ε)) = ρ(R, cos^¯ θ),ρ(R, sin^¯ θ)+ O(ε).

5 Proof of Theorem 1.4

In this section, we will present the k-th order averaged functions up to k = 5 associated to systems (1.4) and (1.5) respectively, and then we use them to prove Theorem1.4.

5.1 Proof of Theorem1.4 (a)

In order to analyze the Hopf bifurcation for system (1.4), applying Theorem2.1, we setγ=2π in (2.2) and we introduce a small parameterεdoing the change of coordinatesx =εX,y=εY.

After that we perform the polar change of coordinatesX=rcosθ,Y=rsinθ, and by doing a